Riemann for Anti-Dummies Part 28

Bringing the Invisible to the Surface

When Carl Friedrich Gauss, writing to his former classmate
Wolfgang Bolyai in 1798, criticized the state of contemporary mathematics for
its "shallowness", he was speaking literally - and, not only about
his time, but also of ours. Then, as now, it had become popular for the academics
to ignore, and even ridicule, any effort to search for universal physical principles,
restricting the province of scientific inquiry to the, seemingly more practical
task, of describing only what's on the surface. Ironically, as Gauss demonstrated
in his 1799 doctoral dissertation on the fundamental theorem of algebra, what's
on the surface, is revealed only if one knows, what's underneath.

Gauss' method was an ancient one, made famous in Plato's metaphor
of the cave, and given new potency by Johannes Kepler's application of Nicholas
of Cusa's method of On Learned Ignorance. For them, the task of the scientist
was to bring into view, the underlying physical principles, that could not be
viewed directly-the unseen that guided the seen.

Take the illustrative case of Pierre de Fermat's discovery
of the principle, that refracted light follows the path of least time, instead
of the path of least distance followed by reflected light. The principle of
least-distance, is a principle that lies on the surface, and can be demonstrated
in the visible domain. On the other hand, the principle of least-time, exists
"behind", so to speak, the visible, brought into view, only in the
mind. On further reflection, it is clear, that the principle of least-time,
was there all along, controlling, invisibly, the principle of least-distance.
In Plato's terms of reference, the principle of least-time is of a "higher
power", than the principle of least-distance.

Fermat's discovery is a useful reference point for grasping
Gauss' concept of the complex domain. As Gauss himself stated, unequivocally,
this is not Leonard Euler's formal, superficial concept of "impossible"
numbers (a fact ignored by virtually all of today's mathematical "experts").
Rather, Gauss' concept of the complex domain, like Fermat's principle of least-time,
brings to the surface, a principle that was there all along, but hidden from
view.

As Gauss emphasized in his jubilee re-working of his 1799 dissertation,
the concept of the complex domain is a "higher domain", independent
of all a priori concepts of space. Yet, it is a domain, "in which one cannot
move without the use of language borrowed from spatial images."

The issue for Gauss, as for Gottfried Leibniz, was to find
a general principle, that characterized what had become known as "algebraic"
magnitudes. These magnitudes, associated initially, with the extension of lines,
squares, and cubes, all fell under Plato's concept of "dunamais",
or "powers".

Leibniz had shown, that while the domain of all "algebraic"
magnitudes consisted of a succession of higher powers, the entire algebraic
domain, was itself dominated by a domain of a still higher power, that Leibniz
called, "transcendental". The relationship of the lower domain of
algebraic magnitudes, to the higher non-algebraic domain of transcendental magnitudes,
is reflected in, what Jacob Bernoulli discovered about the equiangular spiral.

Figure 1

Leibniz and Johann Bernoulli (Jakob's brother) subsequently
demonstrated that his higher, transcendental domain, exists not as a purely
geometric principle, but originates from the physical action of a hanging chain,
whose geometric shape Christaan Huygens called a catenary. (See
Figure 2.) Thus, the physical universe itself demonstrates, that the "algebraic"
magnitudes associated with extension, are not generated by extension. Rather,
the algebraic magnitudes are generated from a physical principle that exists,
beyond simple extension, in the higher, transcendental, domain.

Figure 2

Gauss, in his proofs of the fundamental theorem of algebra,
showed that even though this transcendental physical principle was outside the
visible domain, it nevertheless cast a shadow that could be made visible in
what Gauss called the complex domain.

As indicated in "Gauss' Declaration of Independence,"
the discovery of a general principle for "algebraic" magnitudes was
found, by looking through the "hole" represented by the square roots
of negative numbers, which could appear as solutions to algebraic equations,
but lacked any apparent physical meaning. For example, in the algebraic equation
x2 = 4, "x" signifies the side of a square whose area is 4, while,
in the equation x2 = -4, the "x" signifies the side of a square whose
area is -4, an apparent impossibility. For the first case, it is simple to see,
that a line whose length is 2 would be the side of the square whose area is
4. However, from the standpoint of the algebraic equation, a line whose length
is -2, also produces the desired square.

At first glance, a line whose length is -2 seems as impossible
as a square whose area is -4. Yet, if you draw a square of area 2, you will
see that there are two diagonals, both of which have the power to produce a
new square whose area is 4. These two magnitudes are distinguished from one
another only by their direction, so one is denoted as 2 and the other as -2.

Now extend this investigation to the cube. In the algebraic
equation x3 = 8, there appears to be only one number, 2 which satisfies the equation,
and this number signifies the length of the edge of a cube whose volume is 8.
This appears to be the only solution to this equation since -2x - 2x - 2 = -8.

The anomaly that there are two solutions, which appeared for
the case of a quadratic equation, seems to disappear, in the case of the cube,
for which there appears to be only one solution.

Not so fast. Look at another geometrical problem, that, when
stated in algebraic terms, poses the same paradox--the trisection of an arbitrary
angle. Like the doubling of the cube, Greek geometers could not find a means
for equally trisecting an arbitrary angle, from the principle of circular action
itself. The several methods discovered, (by Archimedes, Erathosthenes, and others),
to find a general principle of trisecting an angle, were similar to those found,
by Plato's collaborators, for doubling the cube. That is, this magnitude could
not be constructed using only a circle and a straight line, but it required
the use of extended circular action, such as conical action.

But, trisecting an arbitrary angle presents another type of
paradox which is not so evident in the problem of doubling the cube. To illustrate
this, make the following experiment:

Draw a circle. For ease of illustration, mark off an angle
of 60 degrees. It is clear that an angle of 20 degrees will trisect this angle
equally. Now add one circular rotation to the 60 degree angle, making an angle
of 420 degrees. It appears these two angles are essentially the same. But, when
420 is divided by 3 we get an angle of 140 degrees. Add another 360 degree rotation
and we get to the angle of 780 degrees, which appears to be exactly the same
as the angles of 60 and 420 degrees. Yet, when we divide 780 degrees by 3 we
get 260 degrees. Keep this up, and you will see that the same pattern is repeated
over and over again. (See Figure 3.)

Figure 3

Looked at from the domain of sense certainty, the angle of
60 degrees can be trisected by only one angle, that is, an angle of 20 degrees.
Yet, when looked at beyond sense certainty, there are clearly three angles that
"solve" the problem.

This illustrates another "hole" in the algebraic
determination of magnitude. In the case of quadratic equations, there seems
to be two solutions to each problem. In some cases, such x2 = -4, those solutions
seem to have a visible existence. While for the case, x2 = -4, there are two solutions,
2√-1 and -2√-1, both of which seem to be "imaginary",
having no physical meaning. In the case of cubic equations, sometimes there
are three visible solutions, such as in the case of trisecting an angle. Yet,
in the case of doubling the cube, there appears to be only one visible solution,
but two "imaginary" solutions, specifically: -1 - √3√-1,
-1 + √3√-1. Biquadratic equations, (for example x4 =
16) , that seem to have no visible meaning themselves, have four solutions,
two "real" (2 and -2) and two "imaginary" (2√-1 and
-2√-1). Things get even more confused for algebraic magnitudes of still
higher powers. This anomaly poses the question that Gauss resolved in his proof
of what he called the fundamental theorem of algebra; that is: how many solutions
are there for any algebraic equation?

The "shallow" minded mathematicians of Gauss' day,
such as Euler, Lagrange, and D'Alembert, took the superficial approach of asserting
that any algebraic equation has as many solutions as it has powers, even if
those solutions were "impossible", such as the square roots of negative
numbers. (This sophist's argument is analogous to saying there is a difference
between man and beast, but, this difference is meaningless.)

Gauss, in his 1799 dissertation, polemically exposed this fraud
for the sophistry it was. "If someone would say a rectilinear equilateral
right triangle is impossible, there will be nobody to deny that. But, if he
intended to consider such an impossible triangle as a new species of triangles
and to apply to it other qualities of triangles, would anyone refrain from laughing?
That would be playing with words, or rather, misusing them."

For, Gauss, no magnitude could be admitted, unless its principle
of generation was demonstrated. For magnitudes associated with the square roots
of negative numbers, that principle was the complex physical action of rotation
combined with extension. Magnitudes generated by this complex action, Gauss
called "complex numbers" in which each complex number denoted a quantity
of combined rotational and extended action. The unit of action in Gauss' complex
domain is a circle, which is one rotation with an extension of unit length.
The number 1 signifies one complete rotation, -1 one half a rotation, √-1 one
fourth a rotation, and -√-1 three fourths a rotation. (See Figure 4.)

Figure 4

These "shadows of shadows", as he called them, were
only a visible reflection of a still higher type of action, that was independent
of all visible concepts of space. These higher forms of action, although invisible,
could nevertheless be brought into view as a projection onto a surface.

Gauss' approach is consistent with that employed by the circles
of Plato's Academy, as indicated by their use of the term "epiphanea"
for surface, which comes from the same root as the word, "epiphany".
The concept indicated by the word "epiphanea" is, " that on which
something is brought into view".

From this standpoint, Gauss demonstrated, in his 1799 dissertation,
that the fundamental principle of generation of any algebraic equation, of no
matter what power, could be brought into view, "epiphanied", so to
speak, as a surface in the complex domain. These surfaces were visible representations,
not, as in the cases of lines, squares and cubes, of what the powers produced,
but of the principle that produced the powers.

To construct these surfaces, Gauss went outside the simple
visible representation of powers, such as squares and cubes, by seeking a more
general form of powers, as exhibited in the equiangular spiral. (See Figure 5.)
Here, the generation of a power, corresponds to the extension
produced by an angular change. For example, the generation of square powers,
corresponds to the extension that results from a doubling of the angle of rotation
around the spiral.

Figure 5

The generation of cubed powers corresponds to the extension
that results from tripling the angle of rotation. Thus, it is the principle
of squaring that produces square magnitudes, and the principle of cubing that
produces cubics. (See figure 6.)

Figure 6

For example, in Figure 7, the complex number z is "squared" when the angle of rotation
is doubled from x to 2x and the length squared from A to A2. In doing
this, the smaller circle maps twice onto the larger "squared" circle.

Figure 7

In Figure 8,
the same principle is illustrated with respect to cubing. Here the angle x is
tripled to 3x, and the length A is cubed to A3. In this case, the smaller circle
maps three times onto the larger, "cubed" circle.

Figure 8

And so on for the higher powers. The fourth power maps the
smaller circle four times onto the larger. The fifth power, five times, and
so forth.

This gives a general principle that determines all algebraic
powers, as, from this standpoint, all powers are reflected by the same action.
The only thing that changes with each power, is the number of times that action
occurs. Thus, each power is distinguished from the others, not by a particular
magnitude, but by a topological characteristic.

In his doctoral dissertation, Gauss used this principle to
generate surfaces that expressed the essential characteristic of powers in an
even more fundamental way. Each rotation and extension, produced a characteristic
right triangle. The vertical leg of that triangle is called the sine and the
horizontal leg of that triangle is called the cosine. (See Figure 9.)

Figure 9

There is a cyclical relationship between the sine and cosine
which is a function of the angle of rotation. When the angle is 0, the sine
is 0 and the cosine is 1. When the angle is 90 degrees the sine is 1 and the
cosine is 0. Looking at this relationship for an entire rotation, the sine goes
from 0 to 1 to 0 to -1 to 0, while the cosine goes from 1 to 0 to -1 to 0 and
back to 1. (See Figure 10)

Figure 10

In Figure 9, as z moves from 0 to 90 degrees, the sine of the
angle varies from 0 to 1, but at the same time, the angle for z2
goes from 0 to 180 degrees, and the sine of z2 varies from 0 to 1
and back to 0. Then as z moves from 90 degrees to 180 degrees, the sine varies
from 1 back to 0, but the angle for z2 has moved from 180 degrees
to 360 degrees, and its sine has varied from 0 to -1 to 0. Thus, in one half
rotation for z, the sine of z2 has varied from 0 to 1 to 0 to -1
to 0.

In his doctoral dissertation, Gauss represented this complex
of actions as a surface. (See Figures 11, 12, 13.) Each point on the surface
is determined so that its height above the flat plane, is equal to the distance
from the center, times the sine of the angle of rotation, as that angle is increased
by the effect of the power. In other words, the power of any point in the flat
plane, is represented by the height of the surface above that point. Thus, as
the numbers on the flat surface move outward from the center, the surface grows
higher according to the power. At the same time, as the numbers rotate around
the center, the sine will pass from positive to negative. Since the numbers
on the surface are the powers of the numbers on the flat plane, the number of
times the sine will change from positive to negative, depends on how much the
power changes the angle (double for square powers, triple for cubics, etc.).
Therefore, each surface will have as many "humps" as the equation
has dimensions. Consequently, a quadratic equation will have two "humps"
up and two "humps" down (Figure 11)

Figure 11

A cubic equation will have three "humps" up and three "humps"
down. (Figure 12). A fourth degree equation four "humps" in each direction, (Figure
13), and so on.

Figure 12

Figure 13

Gauss specified the construction of two surfaces for each algebraic
equation, one based on the variations of the sine and the other based on the
variations of the cosine. (See figures 14a and 14b.)

Figure 14a

Figure 14b

Each of these surfaces will define definite curves where the
surfaces intersect the flat plane. The number of curves will depend on the number
of "humps" which in turn depend on the highest power. Since each of
these surfaces will be rotated 90 degrees to each other, these curves will intersect
each other, and the number of intersections, will correspond to the number of
powers. (See figures 15a and 15b.) If the flat plane is considered
to be 0, these intersections will correspond to the solutions, or "roots"
of the equation. Thus, proving that an algebraic equation has as many roots
as its highest power.

Figure 15a

Figure 15b

Step back and look at this work. These surfaces were produced,
not from visible squares or cubes, but from the general principle of squaring,
cubing, and higher powers. They represent, metaphorically, a principle that
manifests itself physically, but cannot be seen. By projecting this principle,
the general form of Plato's powers, onto these complex surfaces, Gauss has brought
the invisible into view, and made intelligible, something that is incomprehensible
in the superficial world of algebraic formalism.

The effort to make intelligible the implications of the complex
domain was a focus for Gauss throughout his life. Writing to his friend Hansen
on December 11, 1825, Gauss said: "These investigations lead deeply into
many others, I would even say, into the Metaphysics of the theory of space,
and it is only with great difficulty can I tear myself away from the results
that spring from it, as, for example, the true metaphysics of negative and complex
numbers. The true sense of the square root of -1 stands before my mind fully
alive, but it becomes very difficult to put it in words; I am always only able
to give a vague image that floats in the air."